Final answer:
Given an average service time of 1 hour and a standard deviation of 1 hour for AC maintenance, budgeting 1.1 hours per technician may not be sufficient. Calculating the standard error and z-score reveals there is approximately a 79.6% chance of completing service within the budgeted time. For a higher confidence level, more time may need to be allocated.
Step-by-step explanation:
The question asks if budgeting an average of 1.1 hours per technician to service air conditioning units is sufficient based on historical data. The service time has an average (mean) of one hour and a standard deviation of one hour. Since a simple random sample of 70 units will be serviced, we can utilize the Central Limit Theorem to assess if 1.1 hours is enough.
To determine this, we apply the formula for the standard error of the mean (SEM): SEM = σ / √n, where σ is the standard deviation and n is the sample size. In this case, SEM = 1 hour / √70 ≈ 0.12 hours. We are interested in the probability that the sample mean will be less than or equal to the budgeted time of 1.1 hours.
To find this probability, we calculate the z-score: z = (X - μ) / SEM, where X is the budgeted time, and μ is the mean service time. So, z = (1.1 hours - 1 hour) / 0.12 hours ≈ 0.83. Consulting a z-table or using statistical software gives us the probability associated with this z-score. A z-score of 0.83 typically corresponds to a probability of around 79.6%. This means there is approximately a 79.6% chance that a technician will complete the service in 1.1 hours or less.
Considering the probability is less than 95%, which is a common confidence level used in statistics, it may be a little risky to budget only 1.1 hours per technician. If a higher confidence level is desired (e.g., 95% or 99%), then increasing the budgeted time per technician may be advisable.